Abstract
This paper is devoted to the study of a hydrodynamical equation of Riemann type, generalizing the remarkable Gurevich–Zybin system. This multi-component non-homogenous hydrodynamic equation is characterized by the only characteristic flow velocity. The compatible bi-Hamiltonian structures and Lax type representations of the 3-and 4-component generalized Riemann type hydrodynamical system are analyzed. For the first time the obtained results augment the theory of integrability of hydrodynamic type systems, originally developed only for distinct characteristic velocities in homogenous case.
Highlights
Evolution differential equations of special type are capable of describing [1,2,3] many important problems of wave propagation in nonlinear media with distributed parameters, for instance, invisible non-dissipative dark matter, playing a decisive role [4, 5] in the formation of large scale structure in the Universe like galaxies, clusters of galaxies, super-clusters
As follows from the results obtained in this work, the generalized Riemann type hydrodynamical system (10) at N = 2, 3 and N = 4 possesses many infinite hierarchies of conservation laws, both dispersive non-polynomial and dispersionless polynomial ones
This fact can be explained by the fact that the corresponding dynamical systems (16), (58) and (93) possess many, plausibly, infinite sets of algebraically independent compatible implectic structures, which generate via the corresponding gradient like relationships [7, 8] the related infinite hierarchies of conservation laws, and as a by-product, infinite hierarchies of the associated Lax type representations
Summary
Evolution differential equations of special type are capable of describing [1,2,3] many important problems of wave propagation in nonlinear media with distributed parameters, for instance, invisible non-dissipative dark matter, playing a decisive role [4, 5] in the formation of large scale structure in the Universe like galaxies, clusters of galaxies, super-clusters. We remark that, owing to the general symplectic theory results [7,8,9] for nonlinear dynamical systems on smooth functional manifolds, operator (39) defines on the manifold M a closed functional differential two-form. Thereby it is a priori co-implectic (in general, singular symplectic), satisfying on M the standard Jacobi brackets condition. Proposition 4.7 The Riemann type hydrodynamical system (16) is equivalent to a completely integrable bi-Hamiltonian flow on the functional manifold M, allowing the Lax type representation. 2π 0 σ(x; λ)dx, satisfies for all λ ∈ C the gradient relationship (46)
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